Skip to main content

Theoretical BER vs SNR for binary ASK, FSK, and PSK (with MATLAB Code + Simulator)


Bit Error Rate (BER) Equations

In ASK, noise directly affects the signal amplitude, making it the most vulnerable since the data is carried in amplitude changes. In FSK, data is represented by frequency variations, and because noise typically impacts amplitude more than frequency, FSK is more robust than ASK. In PSK, data is encoded in the signal phase, and BPSK specifically uses 180-degree phase shifts, creating the greatest separation between signal points and therefore achieving the lowest bit error rate (BER) for the same power level. BER formulas for ASK, FSK, and PSK modulation schemes.

ASK

BER = 0.5 × erfc(0.5 × √SNR)

FSK

BER = 0.5 × erfc(√(SNR / 2))

PSK

BER = 0.5 × erfc(√SNR)

erfc / Q-function (Click here)

The Mathematical Foundation

The Bit Error Rate is fundamentally derived from the Conditional Probability of Error, where we integrate the Gaussian Noise PDF over the decision region:

\( P_b = \int_{\mathrm{Threshold}}^{\infty} \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-A)^2}{2\sigma^2}} \, dx = Q\!\left(\frac{A}{\sigma}\right) \)

In digital communications, the argument $x$ inside the $Q(x)$ function represents the ratio of the signal's "safety margin" to the noise magnitude. In an AWGN channel, the noise standard deviation is defined as: $\sigma = \sqrt{N_0 / 2}$

BPSK

Distance to boundary is $\sqrt{E_b}$

$x = \frac{\sqrt{E_b}}{\sqrt{N_0/2}}$
$x = \sqrt{\frac{2E_b}{N_0}}$

BFSK

Distance to boundary is $\frac{\sqrt{E_b}}{\sqrt{2}}$

$x = \frac{\sqrt{E_b}/\sqrt{2}}{\sqrt{N_0/2}}$
$x = \sqrt{\frac{E_b}{N_0}}$

BASK

Distance to boundary is $\frac{\sqrt{E_b}}{2}$

$x = \frac{\sqrt{E_b}/2}{\sqrt{N_0/2}}$
$x = \sqrt{\frac{E_b}{2N_0}}$

Why the 0.5 factor?

In 0.5 * erfc(x), the 0.5 represents the 50% probability of transmitting a '0' vs a '1' in a random bitstream.

Eb/N0 vs SNR

While SNR is the power ratio, Eb/N0 is the energy per bit, allowing us to compare schemes with different bandwidths fairly.

Theoretical BER vs SNR for Amplitude Shift Keying (ASK)

The theoretical Bit Error Rate (BER) for binary ASK depends on how binary bits are mapped to signal amplitudes. For typical cases:

If bits are mapped to 1 and -1, the BER is:

BER = Q(√(2 × SNR))

If bits are mapped to 0 and 1, the BER becomes:

BER = Q(√(SNR / 2))

Where:

  • Q(x) is the Q-function: Q(x) = 0.5 × erfc(x / √2)
  • SNR: Signal-to-Noise Ratio
  • N₀: Noise Power Spectral Density

Understanding the Q-Function and BER for ASK

  • Bit '0' transmits noise only
  • Bit '1' transmits signal (1 + noise)
  • Receiver decision threshold is 0.5

BER is given by:

Pb = Q(0.5 / σ), where σ = √(N₀ / 2)

Using SNR = (0.5)² / N₀, we get:

BER = Q(√(SNR / 2))

ASK BER Formula Derivation

Theoretical BER vs SNR for Frequency Shift Keying (FSK)

For binary FSK, the theoretical BER is:

BER = Q(√(SNR))

BER vs SNR for FSK

The Q-function is defined as:

Q(x) = 0.5 × erfc(x / √2)

BER Formula for BFSK

Similarities Between ASK and FSK

  • Both BERs decrease as SNR increases
  • Both use the Q-function for analytical BER calculation
  • FSK generally performs better under noisy conditions

MATLAB Code for Theoretical BER vs SNR

Binary ASK (BASK)

% The code is written by SalimWireless.Com 

clc;
clear all;
close all;

SNRdB = 0:20; 
SNR = 10.^(SNRdB/10); 

BER_th = (1/2) * erfc(0.5 * sqrt(SNR));

semilogy(SNRdB, BER_th, '-rh', 'linewidth', 2.5);
grid on;
title('Theoretical Bit Error Rate vs. SNR for Binary ASK Modulation');
xlabel('SNR (dB)');
ylabel('BER');
legend('Theoretical');
axis([0 20 1e-5 1]);

Binary FSK (BFSK)

% The code is written by SalimWireless.Com 

clc;
clear;
close all;

SNRdB = 0:1:10;              
SNR = 10.^(SNRdB/10);        

BER_th = (1/2) * erfc(sqrt(SNR / 2));

disp('SNR (dB)    Theoretical BER');
disp([SNRdB', BER_th']);

figure;
semilogy(SNRdB, BER_th, '-kh', 'LineWidth', 2);
xlabel('SNR (dB)');
ylabel('Bit Error Rate (BER)');
title('Theoretical BER vs SNR for BFSK');
grid on;

Theoretical BER vs SNR for BPSK -> (Click Here)

BER vs SNR Simulation



Comparison of ASK, FSK, and PSK Performance

Feature ASK (OOK) BFSK BPSK
Power Efficiency Low Medium High
Bandwidth Efficiency High Low High
Noise Immunity Poor (Sensitive to Amp) Good Excellent
Best Used In Fiber Optics, RFID Caller ID, Paging Deep Space, Satellite

Read More

Key Terms Explained

Eb/N0 (Energy per Bit to Noise Power Density):
The normalized SNR measure, also known as the "SNR per bit." It is the most important metric for comparing different modulation schemes regardless of bandwidth. [Read More] 
Q-Function:
The Q-function represents the tail probability of the standard normal distribution. In BER terms, it tells us the probability that noise will exceed the decision boundary. Read more
Coherent vs. Non-Coherent Detection:
Coherent detection (used in the formulas above) requires the receiver to be in phase-sync with the transmitter. Non-coherent detection is easier to build but results in a higher BER.
Impact of AWGN on BER vs SNR of FSK and PSK →

Further Reading

  1. BER vs SNR for ASK, FSK, and PSK with Online Simulator 
  2. Understanding the Q-function in BASK, BFSK, and BPSK
  3. BER for M-ary PSK and QAM
  4. Constellation Diagrams
📊 MATLAB & ONLINE SIMULATIONS (Web-Based)
🎓 GATE & UGC - NET SOLUTIONS
📡 WIRELESS & SIGNAL PROCESSING


Contact Us

Name

Email *

Message *

Popular Posts

MATLAB Code for Constellation Diagram of QAM configurations such as 4, 8, 16, 32, 64, 128, and 256-QAM

📘 Overview of QAM 🧮 4-QAM MATLAB 🧮 16-QAM MATLAB 🚀 Online Simulator 📂 Other Topics on Constellation Diagrams... ▼ 🧮 MATLAB Code for 4-QAM 🧮 MATLAB Code for 16-QAM 🧮 MATLAB Code for m-ary QAM 🧮 Simulator for m-ary PSK 🧮 Simulator for m-ary QAM 🧮 Overview of Energy per Bit (Eb / N0) 🧮 Simulator for ASK, FSK, and PSK Overview of QAM One of the best-performing modulation techniques is QAM [↗] . Here, we modulate the symbols by varying the carrier signal's amplitude and phase in response to the variation in the message signal (or voltage variation). So, we may say that QAM is a combination of phase and amplitude modulation. Additionally, it performs better than ASK or PSK [↗] . In fact, any constellation for any type of modulatio...
Read more

UGC NET Electronic Science Previous Year Question Papers with Solutions

Download Papers and Solutions Exam Pattern Preparation Tips FAQs More Home / Engineering & Other Exams / UGC NET 2026 PYQ 📊 Exam Highlights: Electronic Science (88) Feature Details Junior Research Fellowship (JRF) ₹37,000 + HRA per month Eligibility M.Sc/M.Tech in Electronics (55%) Validity of Certificate JRF (3 Years) | Lectureship (Lifetime) 📥 Download UGC NET Electronics PDFs Complete collection of previous year question papers, answer keys and explanations for Subject Code 88. Start Downloading 📂 View All Question Papers June 2026 - Question Paper Download PDF June 202...
Read more

GMSK BER vs SNR (with Simulation)

  GMSK Performance: Why the BER vs. SNR Curve Matters If you’ve ever wondered why GMSK (Gaussian Minimum Shift Keying) became the backbone of the global GSM revolution, the answer lies in its unique balance of spectral efficiency and power performance. But there is no free lunch in engineering. To get that sleek, narrow spectrum, you have to pay a price in Bit Error Rate (BER) . In this guide, we break down the theoretical BER vs. SNR performance of GMSK, the "magic" of the BT product, and how to optimize your link budget for real-world conditions. What is GMSK and Why Should You Care? GMSK is a derivative of MSK (Minimum Shift Keying) that uses a Gaussian pre-modulation filter. This filter "smooths" the phase transitions, significantly reducing sideband power. The result? A constant envelope signal that allows you to use high-efficiency Class-C amplifiers without distorting the signal. This is the secr...
Read more

Flat vs Frequency Selective Online Simulator

Flat vs Frequency Selective Online Simulator Channel Type Without Fading Flat Fading Multipaths Nakagami m SNR(dB) Run Simulation Input Signal Signal After Fading Constellation Diagram BER vs SNR Explore Advanced Flat vs Frequency-Selective Fading Simulator Want to see these equations in action? Visualize it. Launch Simulator Tool Flat Fading and Frequency-Selective Fading Effects of Multipath Components and the Nakagami-m Parameter 1. Flat Fading Flat fading occurs when the signal bandwidth is smaller than the channel's coherence bandwidth, meaning that all frequency components of the transmitted signal experience approximately the same fading. It is typically associated with narrowband signals, where the multipath delay spread is small relative to the symbol duration. ...
Read more

Orthogonal Time Frequency Space (OTFS) (with MATLAB)

In OTFS (Orthogonal Time Frequency Space) modulation — a scheme designed for high-Doppler and time-varying wireless channels — the terms ISFFT and SFFT are key mathematical transformations used to move between different representation domains. Figure: OTFS block diagram 1. ISFFT — Inverse Symplectic Finite Fourier Transform Purpose: Transforms data symbols from the delay-Doppler domain to the time-frequency domain . \[ X[n, m] = \frac{1}{\sqrt{NM}} \sum_{k=0}^{N-1} \sum_{l=0}^{M-1} x[k, l] \, e^{j2\pi \left( \frac{nk}{N} - \frac{ml}{M} \right)} \] Here, \( N \) is the number of Doppler bins (time slots), and \( M \) is the number of delay bins (subcarriers). The ISFFT maps each data symbol from the delay-Doppler grid (where the channel is sparse and easier to equalize) to the time-frequency grid (where standard multicarrier modulation like OFDM can be applied). 2. SFFT — Symplectic Finite Fourier Transform Purpose: Performs the reverse operation ...
Read more

Quantization Signal to Noise Ratio (Q-SNR)

What is Quantization? Quantization: Quantization in Digital Signal Processing (DSP) is the process of mapping a large set of input values to a smaller set. It’s essential for converting analog signals into digital form. × Quantization Explanation For a signal varies from -8 V to +8 V, giving a total quantization range of 16 V. If the number of quantization levels is 4, the step size will be: \[ v_{\min} = -8, \quad v_{\max} = 8, \quad L = 4 \] Quantization step size: \[ \Delta = \frac{v_{\max} - v_{\min}}{L} = \frac{8 - (-8)}{4} = \frac{16}{4} = 4 \] Partition boundaries (decision levels): \[ p_0 = -8, \quad p_1 = -8 + 4 = -4, \quad p_2 = 0, \quad p_3 = 4, \quad p_4 = 8 \] Quantization codebook (reconstruction levels): \[ c_i = v_{\min} + \left(i + \frac{1}{2}\right) \Delta, \quad i = 0, 1, 2, 3 \] Calculate each codeword: \[ c_0 = -8 + \left(0 + \frac{...
Read more

Design of CMOS XOR/XNOR Gates

Design of CMOS XOR/XNOR Gates The semiconductor industry has experienced rapid integration of multimedia applications into mobile electronics, leading to very high integration density in CMOS VLSI. As operating frequencies increase, power consumption, speed, silicon area, and reliability become critical considerations. The XOR-XNOR circuits are fundamental building blocks in arithmetic circuits (Full Adders, Multipliers), compressors, comparators, parity checkers, code converters, error-detecting/correcting codes, and phase detectors. Their performance directly impacts the complex circuits they are used in. Design goals include full output voltage swing, low power consumption, reduced transistor count, minimal delay, and simultaneous non-skewed outputs. Static Logic (Static CMOS) Stat...
Read more

Random Variable vs. Random Process: Definitions, Examples, and Key Differences

Random Variable vs. Random Process: Definitions, Examples, and Key Differences Random Variable and Random Process In the fields of probability theory and statistics, understanding the distinction between a Random Variable and a Random Process (Stochastic Process) is fundamental. This guide breaks down these concepts with mathematical precision and practical examples. 1. Random Variable (RV) Definition: A Random Variable is a deterministic function that maps each possible outcome of a random experiment to a unique real number. Mathematical Representation A random variable X is defined as a mapping from the sample space S to the set of real numbers ℝ : X : S → ℝ S (Sample Space): The set of all possible outcomes of an experiment. s ∈ S: A specific outcome. X(s): The numerical value asso...
Read more